

where \(\bar{a} \triangleq \sum _{k\in \mathcal {K}_\mathrm {r}}\sum _{j=1}^{K}\left|\tau _k\right|^2\mathbf {w}_{j}^H\mathbf {h}_{\mathrm {d},k}\mathbf {h}_{\mathrm {d},k}^H\mathbf {w}_{j}\)  is a constant independent of \(\phi _{\mathrm {r}}\) . It can be observed that, although the objective function in (REF ) is continuous and convex, problem (REF ) is still difficult to solve due to the constant modulus constraint (). There are two popular methods for handling this type of constraint: non-convex relaxation and alternating minimization. However, the non-convex relaxation method always suffers a performance loss and the alternating minimization method usually has slow convergence as the number of variables increases. To effectively solve this problem, we adopt the Riemannian-manifold method [1]} for directly solving problem (REF ) with very fast convergence [2]}.
The search space in problem (REF ) can be regarded as the product of \(M\)  complex circles, which can be given by
\(\mathcal {C}^{M} \triangleq \left\lbrace \mathbf {x} \in \mathbb {C}^{M}:\left|{x}_l\right|=1, l=1,2, \cdots , M\right\rbrace ,\) 
